# American Institute of Mathematical Sciences

August  2017, 11(4): 601-621. doi: 10.3934/ipi.2017028

## Landmark-guided elastic shape analysis of human character motions

 1 Florida State University, Department of Mathematics, 208 Love Building, 1017 Academic Way, Tallahassee, FL 32306-4510, USA 2 Department of Mathematical Sciences, NTNU -Norwegian University of Science and Technology, 7491 Trondheim, Norway

* Corresponding author: Martin Bauer

Received  December 2014 Revised  March 2017 Published  June 2017

Fund Project: MB was supported by FWF project P24625

Motions of virtual characters in movies or video games are typically generated by recording actors using motion capturing methods. Animations generated this way often need postprocessing, such as improving the periodicity of cyclic animations or generating entirely new motions by interpolation of existing ones. Furthermore, search and classification of recorded motions becomes more and more important as the amount of recorded motion data grows.

In this paper, we will apply methods from shape analysis to the processing of animations. More precisely, we will use the by now classical elastic metric model used in shape matching, and extend it by incorporating additional inexact feature point information, which leads to an improved temporal alignment of different animations.

Citation: Martin Bauer, Markus Eslitzbichler, Markus Grasmair. Landmark-guided elastic shape analysis of human character motions. Inverse Problems & Imaging, 2017, 11 (4) : 601-621. doi: 10.3934/ipi.2017028
 [1] M. F. Abdelkader, W. Abd-Almageed, A. Srivastava and R. Chellappa, Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455. doi: 10.1016/j.cviu.2010.10.006. Google Scholar [2] D. Alekseevsky, A. Kriegl, M. Losik and P. W. Michor, The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276. Google Scholar [3] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015.Google Scholar [4] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9. doi: 10.5244/C.29.DIFFCV.9. Google Scholar [5] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73. doi: 10.1137/16M1066282. Google Scholar [6] M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165. doi: 10.1016/j.difgeo.2014.04.008. Google Scholar [7] M. Bauer, M. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97. doi: 10.1007/s10851-013-0490-z. Google Scholar [8] M. Bauer, M. Bruveris and P. W. Michor, R-transforms for Sobolev H2-metrics on spaces of plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56. doi: 10.4310/GIC.2014.v1.n1.a1. Google Scholar [9] M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436. doi: 10.1016/j.difgeo.2014.12.008. Google Scholar [10] A. Bruderlin and L. Williams, Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104. doi: 10.1145/218380.218421. Google Scholar [11] M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp. doi: 10.1017/fms.2014.19. Google Scholar [12] Carnegie-Mellon, Carnegie-mellon mocap database, 2003.Google Scholar [13] E. Celledoni, M. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304. doi: 10.3934/jgm.2016008. Google Scholar [14] M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190. doi: 10.1007/s00371-014-1001-y. Google Scholar [15] M. Fuchs, B. Jüttler, O. Scherzer and H. Yang, Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102. doi: 10.1007/s10851-009-0156-z. Google Scholar [16] L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224.Google Scholar [17] L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568. doi: 10.1145/1186562.1015760. Google Scholar [18] S. Kurtek, A. Srivastava, E. Klassen and H. Laga, Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438. doi: 10.1111/cgf.12063. Google Scholar [19] H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52. doi: 10.1016/j.jtbi.2014.07.036. Google Scholar [20] W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011. Google Scholar [21] A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445. doi: 10.4171/IFB/196. Google Scholar [22] P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004. Google Scholar [23] W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15. doi: 10.1109/CVPR.2004.1315138. Google Scholar [24] W. Mio, A. Srivastava and S. Joshi, On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324. doi: 10.1007/s11263-006-9968-0. Google Scholar [25] T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226. doi: 10.1111/j.1467-8659.2009.01591.x. Google Scholar [26] C. Samir, P.-A. Absil, A. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73. doi: 10.1007/s10208-011-9091-7. Google Scholar [27] J. Shah, $H^0$-$type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137. doi: 10.1090/S0033-569X-07-01084-4. Google Scholar [28] J. Shah, An$H^2$Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506. doi: 10.1016/j.aam.2013.06.003. Google Scholar [29] E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75. doi: 10.1109/CVPR.2004.1315185. Google Scholar [30] A. Srivastava, E. Klassen, S. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428. doi: 10.1109/TPAMI.2010.184. Google Scholar [31] J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552. doi: 10.1214/13-AOAS701. Google Scholar [32] Q. Xie, S. Kurtek, E. Klassen and A. Srivastava, Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929. doi: 10.1214/14-EJS938D. Google Scholar [33] L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic). doi: 10.1137/S0036139995287685. Google Scholar [34] L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57. doi: 10.4171/RLM/506. Google Scholar show all references ##### References:  [1] M. F. Abdelkader, W. Abd-Almageed, A. Srivastava and R. Chellappa, Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455. doi: 10.1016/j.cviu.2010.10.006. Google Scholar [2] D. Alekseevsky, A. Kriegl, M. Losik and P. W. Michor, The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276. Google Scholar [3] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015.Google Scholar [4] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9. doi: 10.5244/C.29.DIFFCV.9. Google Scholar [5] M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73. doi: 10.1137/16M1066282. Google Scholar [6] M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165. doi: 10.1016/j.difgeo.2014.04.008. Google Scholar [7] M. Bauer, M. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97. doi: 10.1007/s10851-013-0490-z. Google Scholar [8] M. Bauer, M. Bruveris and P. W. Michor, R-transforms for Sobolev H2-metrics on spaces of plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56. doi: 10.4310/GIC.2014.v1.n1.a1. Google Scholar [9] M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436. doi: 10.1016/j.difgeo.2014.12.008. Google Scholar [10] A. Bruderlin and L. Williams, Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104. doi: 10.1145/218380.218421. Google Scholar [11] M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp. doi: 10.1017/fms.2014.19. Google Scholar [12] Carnegie-Mellon, Carnegie-mellon mocap database, 2003.Google Scholar [13] E. Celledoni, M. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304. doi: 10.3934/jgm.2016008. Google Scholar [14] M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190. doi: 10.1007/s00371-014-1001-y. Google Scholar [15] M. Fuchs, B. Jüttler, O. Scherzer and H. Yang, Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102. doi: 10.1007/s10851-009-0156-z. Google Scholar [16] L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224.Google Scholar [17] L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568. doi: 10.1145/1186562.1015760. Google Scholar [18] S. Kurtek, A. Srivastava, E. Klassen and H. Laga, Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438. doi: 10.1111/cgf.12063. Google Scholar [19] H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52. doi: 10.1016/j.jtbi.2014.07.036. Google Scholar [20] W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011. Google Scholar [21] A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445. doi: 10.4171/IFB/196. Google Scholar [22] P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004. Google Scholar [23] W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15. doi: 10.1109/CVPR.2004.1315138. Google Scholar [24] W. Mio, A. Srivastava and S. Joshi, On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324. doi: 10.1007/s11263-006-9968-0. Google Scholar [25] T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226. doi: 10.1111/j.1467-8659.2009.01591.x. Google Scholar [26] C. Samir, P.-A. Absil, A. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73. doi: 10.1007/s10208-011-9091-7. Google Scholar [27] J. Shah,$H^0$-$ type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137. doi: 10.1090/S0033-569X-07-01084-4. Google Scholar [28] J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506. doi: 10.1016/j.aam.2013.06.003. Google Scholar [29] E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75. doi: 10.1109/CVPR.2004.1315185. Google Scholar [30] A. Srivastava, E. Klassen, S. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428. doi: 10.1109/TPAMI.2010.184. Google Scholar [31] J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552. doi: 10.1214/13-AOAS701. Google Scholar [32] Q. Xie, S. Kurtek, E. Klassen and A. Srivastava, Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929. doi: 10.1214/14-EJS938D. Google Scholar [33] L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic). doi: 10.1137/S0036139995287685. Google Scholar [34] L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57. doi: 10.4171/RLM/506. Google Scholar
The dynamic programming algorithm can be sped up massively by only considering predecessor indices $(k, l)$ close to the current index $(i, j)$ in (29)
This skeleton, which is based on data from the CMU Graphics Lab Motion Capture Database [12], was used for the animation experiments. Figure taken from [14]
Effect of picking different feature points when matching two shapes. The top left figure shows results for shape matching using only an elastic energy functional without feature points. The remaining figures show matching results for different combinations of feature points. Corresponding markers on the left and right are matched, resulting in different paths between the given curves
Effect of picking different feature points when matching two different hand shapes. In the top row, no feature points were set. The purely elastic matching produces distorted shapes along the geodesic path between the two hand shapes. In the middle row, feature points were set to match the tips of ring and index fingers correspondingly. This results in more natural interpolated shapes. In the bottom row, we see how incorrect feature matches cause some fingers to merge and new fingers to grow along the interpolation between the two shapes. Corresponding markers on the left and right are matched, resulting in different paths between the given curves. Colors along the curves indicate parametrization
Example of using various methods to interpolate between two walking animations. The blue and orange lines are the trajectories of the left and right feet respectively. Note in particular how the two walking animations have different numbers of steps and how the various interpolated animations struggle with that. We have from left to right and top to bottom the following methods: linear interpolation of the Euler angles, elastic matching without reparametrization, elastic matching with reparametrization and finally elastic and feature point matching with reparametrization
Example of using various methods to interpolate between two walking animations stepping over an obstacle. The blue and orange lines are the trajectories of the left and right feet respectively. We have from left to right and top to bottom the following methods: linear interpolation of the Euler angles, elastic matching without reparametrization, elastic matching with reparametrization and finally elastic and feature point matching with reparametrization
Computation times for Fig. 4
 No. of points Elastic reparam. Correct features Incorrect features 100 17 s (6) 11 s (3) 22 s (6) 200 32 s (6) 22 s (3) 50 s (6) 400 67 s (6) 48 s (3) 95 s (8)
 No. of points Elastic reparam. Correct features Incorrect features 100 17 s (6) 11 s (3) 22 s (6) 200 32 s (6) 22 s (3) 50 s (6) 400 67 s (6) 48 s (3) 95 s (8)
 [1] Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008 [2] Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011 [3] Jian Zhao, Fang Deng, Jian Jia, Chunmeng Wu, Haibo Li, Yuan Shi, Shunli Zhang. A new face feature point matrix based on geometric features and illumination models for facial attraction analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1065-1072. doi: 10.3934/dcdss.2019073 [4] Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961 [5] Daniela Calvetti, Jenni Heino, Erkki Somersalo, Knarik Tunyan. Bayesian stationary state flux balance analysis for a skeletal muscle metabolic model. Inverse Problems & Imaging, 2007, 1 (2) : 247-263. doi: 10.3934/ipi.2007.1.247 [6] Srdjan Stojanovic. Interest rates risk-premium and shape of the yield curve. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1603-1615. doi: 10.3934/dcdsb.2016013 [7] James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013 [8] Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885 [9] Roberto Avanzi, Nicolas Thériault. A filtering method for the hyperelliptic curve index calculus and its analysis. Advances in Mathematics of Communications, 2010, 4 (2) : 189-213. doi: 10.3934/amc.2010.4.189 [10] Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29 [11] Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419 [12] P. Cerejeiras, M. Ferreira, U. Kähler, F. Sommen. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Communications on Pure & Applied Analysis, 2007, 6 (3) : 619-641. doi: 10.3934/cpaa.2007.6.619 [13] Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453 [14] Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457 [15] C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457 [16] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [17] Urszula Foryś, Yuri Kheifetz, Yuri Kogan. Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 511-525. doi: 10.3934/mbe.2005.2.511 [18] Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013 [19] Santanu Sarkar, Subhamoy Maitra. Some applications of lattice based root finding techniques. Advances in Mathematics of Communications, 2010, 4 (4) : 519-531. doi: 10.3934/amc.2010.4.519 [20] Santiago Moral, Victor Chapela, Regino Criado, Ángel Pérez, Miguel Romance. Efficient algorithms for estimating loss of information in a complex network: Applications to intentional risk analysis. Networks & Heterogeneous Media, 2015, 10 (1) : 195-208. doi: 10.3934/nhm.2015.10.195

2018 Impact Factor: 1.469

## Metrics

• HTML views (16)
• Cited by (1)

• on AIMS